Pseudorandom Generators for Combinatorial Checkerboards

We define a combinatorial checkerboard to be a function f:{1, ⋯, m}^d → {1, -1} of the form f(u₁, ⋯, u_d) = Π_i=1^d f_i(u_i) for some functions f_i:{1, ⋯, m} → {1, -1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1, -1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m=2. We construct a pseudorandom generator that ϵ-fools all combinatorial checkerboards with seed length O(log m + log d·log log d + log^3/2 1/ϵ). Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length O(log m + log² d + log d·log 1/ϵ). Our seed length is better except when 1/ϵ ≥ d^{ω(log d)}.